Problem: A function $f$ from the integers to the integers is defined as follows:
\[f(n) = \left\{
\begin{array}{cl}
n + 3 & \text{if $n$ is odd}, \\
n/2 & \text{if $n$ is even}.
\end{array}
\right.\]Suppose $k$ is odd and $f(f(f(k))) = 27.$  Find $k.$
Solution: Since $k$ is odd, $f(k) = k + 3.$  Then $k + 3$ is even, so
\[f(k + 3) = \frac{k + 3}{2}.\]If $\frac{k + 3}{2}$ is odd, then
\[f \left( \frac{k + 3}{2} \right) = \frac{k + 3}{2} + 3 = 27.\]This leads to $k = 45.$  But $f(f(f(45))) = f(f(48)) = f(24) = 12,$ so $\frac{k + 3}{2}$ must be even.  Then
\[f \left( \frac{k + 3}{2} \right) = \frac{k + 3}{4} = 27.\]This leads to $k = 105.$  Checking, we find $f(f(f(105))) = f(f(108)) = f(54) = 27.$

Therefore, $k = \boxed{105}.$